PINNs for Phase Space Flows

Based on the work of Raissi et al., we expand on the paradigm of Physics-Informed Neural Networks. Specifically, we work with the discrete time scheme (which uses high order Implicit Runge Kutta) and generalize it to higher order dimensional input and output spaces. Here, we test the following systems: 1D Allen-Cahn, 1D Nonlinear Schrödinger Equation (2 output dimensions at it is a complex field), 2D heat equation (2D input, 1D output), and fluid mechanics with the Taylor Green Vortex (2D input, 2D output). 

We then proceed to expand on the use of PINNs, as we use them to solve for particle phase space prediction (in discrete time). We work with both a charged particle in a non-uniform electric field, and a body being acted upon by a central force.